2048

In a perfect game where only 2's spawn, when its GAME OVER, the $1^{st}$ tile would be 2 ( $2^1$ ), $2^{nd}$ tile would be 4 ( $2^2$ ), and similarly, the $n^{th}$ tile would be $2^n$ .Since, in the game there are only 16 tiles, the maximum achievable score will be $2^{16} = 65536$